You are here

Geometry Through History: Euclidean, Hyperbolic, and Projective Geometries

Meighan I. Dillon
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
, on

This book has its origins in a geometry course that was developed and taught by the author to prospective secondary school teachers, but which apparently came to be populated by other kinds of students (for example, physics and computer science majors) as well. The title may be a little misleading: although there is a lot of interesting geometry to be found here, and although the chapters follow an historical path and contain some historical content, the history is not developed as much as one might expect it to be, with a number of omissions that struck me as puzzling.

There are ten chapters. The first six comprise a course in what I would call the foundations of geometry. In more detail: the first chapter looks at the first book of Euclid’s Elements, reproducing proofs in somewhat more modern language, and also giving some indication of how Euclid made logical errors in reasoning that, by current standards of mathematical reasoning, do not stand up to scrutiny. The second then focuses on “neutral” geometry, which is the geometry we get if we assume all the axioms of Euclidean geometry except the famous fifth postulate; the idea here is to see what theorems of Euclidean geometry can be proved without the fifth postulate, and what changes to other theorems of Euclidean geometry are required. (For example, although it can no longer be proved that the sum of the angles of a triangle is 180 degrees, one can prove that the angle sum is less than or equal to this number.) In chapter 3, the assumption of neutrality regarding the parallel postulate is dropped and the hyperbolic parallel postulate, asserting that through a given point not on a given line, there are at least two lines parallel to the line, is adopted; the chapter then surveys some of the basic results of this non-Euclidean geometry.

Chapter 4 sketches a rigorous development of Euclidean (solid) geometry via the axioms used by Hilbert in his famous Foundations of Geometry. The exposition here is fairly streamlined (the entire chapter is about 25 pages long), and a number of results are left to the reader as exercises. Chapter 5 then returns to Euclidean geometry, going beyond Book I of the Elements to discuss topics discussed in later books, and also some topics not discussed in the Elements at all. One of the topics covered here is inversion in a circle, which is in turn used in chapter 6, which describes the Poincaré disc model of hyperbolic geometry. Other topics discussed in chapter 5 include the theorems of Ceva and Menelaus, and the nine-point circle.

Although there is no chapter specifically devoted to axiom systems and their models, these topics are introduced “on the fly”, so to speak, especially in chapters 4 and 6. The discussion of these topics is fairly brief, perhaps because the author wished to avoid getting bogged down in these issues. Whether this approach is optimal is likely one of personal taste; I tend to think a more systematic discussion might have been advisable. Specifically, I think the use of models in chapter 4 would have enhanced that chapter, allowing the reader to see concrete interpretations of the various axioms discussed therein.

The remaining four chapters of the book shift gears somewhat, and cover a selection of topics in geometry that are all linked to abstract or linear algebra. Chapter 7 is on affine geometry, which, as done here, refers to the underlying geometry of a vector space: a point is an element of the space, and a line is a coset of a one-dimensional subspace. If we start, for example, with a two-dimensional space over the field \(\mathbb{R}\) of real numbers, we obtain the real affine plane, and can give elegant algebraic proofs of many theorems of Euclidean geometry that rely only on incidence alone, rather than on distance or angles. The author works in greater generality, however, not restricting herself to two-dimensional spaces, and also working not just over \(\mathbb{R}\) but over arbitrary fields.

Chapter 8 is on projective geometry. The approach to this subject is linear algebraic as well, and builds on the material of the preceding chapter on affine spaces defined by a field (or more precisely a vector space over that field). Dillon begins with affine three-space defined by a field F (i.e., a three-dimensional vector space over F), and adds “points at infinity” to form projective three-space. Then she shifts gears and gives some axioms for both projective space and the projective plane. Subsequent sections in the chapter touch on various additional topics, such as the theorems of Desargues and Pappus, harmonic sequences and homogenous coordinates for the projective plane defined by a field.

Chapter 9, on algebraic curves, gives an introduction to some of the ideas of algebraic geometry in affine and projective spaces, culminating in a statement of Bézout’s theorem, and a rough sketch of how it is proved.

Finally, chapter 10 discusses rotations of the plane and three-dimensional Euclidean space. Quaternions are introduced late in the chapter and their relationship to rotations addressed.

This is quite a lot of material — more, I think, than can be covered in one semester. A dependence chart, or some indication of possible syllabi for courses, would have been helpful. The first six chapters seem to form, as previously mentioned, a fairly cohesive unit, but it would be helpful for a perspective reader to know just what material from these chapters is necessary for study of the remaining four.

I thought that the last four chapters were somewhat less successful than the six that preceded them. For one thing, these chapters necessarily invest a fair amount of time in developing the relevant algebraic background. The chapter on affine geometry, for example, spends more time discussing background linear and abstract algebra than it does discussing geometry. Likewise, the chapter on algebraic curves spends a great deal of time discussing algebraic preliminaries (integral domains, UFDs, polynomials, resultants), and we only really get to geometry fairly late in the chapter. By the time the algebraic preliminaries in both of these chapters have been disposed of, the amount of time and space available to do geometry is limited; this all struck me as a lot of preliminary time spent leading up to an inadequate payoff.

The author’s chapter on projective geometry essentially makes it dependent on the previous chapter on affine geometry, and therefore makes it hard to introduce this subject directly after the first six chapters on Euclidean and hyperbolic geometry. Yet many instructors, having just discussed hyperbolic geometry (where there are lots of lines that are parallel to a given line and which pass through a given point not on that line) might wish to discuss a geometry where there are no such parallel lines.

In addition, the transition from the first six chapters to the last four seemed to me a bit abrupt, particularly since there are other topics that would fit in better. Particularly in a class with a number of prospective secondary school teachers in the audience, a more extended discussion of compass and straightedge constructions than the fairly brief account given here (particularly emphasizing the impossible ones) would be very valuable, as would a chapter on general geometric transformations (classifying plane isometries, for example, and using them to solve geometric problems, particularly with regard to symmetry; this topic would be of value to the physics majors in the class as well).

I mentioned in the first paragraph of this review that I thought there were some puzzling omissions in the historical discussions. The lack of any treatment of the three unsolvable compass and straightedge constructions (trisecting the angle, doubling the cube, squaring the circle) is one of them, as is the omission of any discussion of the constructability of the regular n-gon, but there are others.

For one thing, geometry in this book begins with Euclid, which necessarily means that there is no discussion of pre-Greek geometry. This decision can be defended on the grounds that modern mathematics began with the Greeks, but it still struck me as odd that something like Plimpton 322 is not even mentioned, especially since the Pythagorean theorem is, and there is a connection between the two. And I find it even harder to believe that a book that contains the words “geometry” and “history” in its title would omit any reference to Thales.

In addition, given the fact that the Pythagorean theorem is discussed, I was surprised to see no discussion of the fact that \(\sqrt2\), the length of the hypotenuse of a right triangle with both legs having unit length, is irrational, and the significance of results like this for the ancient Greeks.

Another Greek mathematician that receives shorter shrift than one might expect is Archimedes. He is mentioned in connection with the Archimedean axiom, but there is little attention given to some of his geometrical accomplishments.

There is little discussion of the historical roots of projective geometry. The book’s introduction of projective planes and spaces in terms of arbitrary fields may lead a student to mistakenly assume that this was the way the subject arose historically, when in fact the term “projective plane” historically referred only to the extended Euclidean plane (i.e., the real number field, not arbitrary fields) and was based on problems in perspective drawing that arose during the Renaissance. The relationship between perspective drawing and projective geometry is only briefly mentioned in the book.

There is a beautiful connection between coordinatization and the theorems of Pappus and Desargues: a projective plane, defined axiomatically, can be coordinatized by a division ring if and only if Desargues’ theorem holds in that plane, and can be coordinatized by a field if and only if Pappus’ theorem does. Unfortunately, this topic is also not discussed in the chapter on projective geometry.

I also thought that the author’s account of the history of hyperbolic geometry leaves many interesting things unsaid. Dillon summarizes the history in a few pages but omits much of the details that students find interesting. Although she says, for example, that attempts were made to prove Euclid’s Fifth Postulate but were ultimately all found to be faulty, she doesn’t give examples of these faulty proofs and explain why they were faulty. She also omits much of the human interest involved in the development of the subjects: the letters between Wolfgang and Janos Bolyai, for example, are fascinating, as is the correspondence between the elder Bolyai and Gauss, but they are not discussed or quoted here. Greenberg’s Euclidean and Non-Euclidean Geometries: Development and History, by contrast, gives a much more compelling account of how people gradually came to realize that hyperbolic geometry was a logically consistent area of mathematics in its own right.

I should make it clear that I am not suggesting that any book on geometry must contain these topics. It seems to me, however, that when you title a book Geometry Through History, you raise certain expectations, and I am not sure these are fully met by this text. Books with similar titles (Geometry by Its History by Ostermann and Wanner and Revolutions of Geometry by O’Leary, for example), do seem to put more emphasis on history than does this text. And, of course, for a really detailed look at the history of geometry, there is 5000 Years of Geometry by Scriba and Schreiber and, for 19th century geometry, Worlds Out of Nothing by Jeremy Gray. Perhaps a title like “Topics in Geometry” or “Introduction to Geometry” would have been more accurate.

But there are worse things to say about a book than the fact that it may not have a fully accurate title. There is, in fact, a lot to like about this book. It discusses a lot of interesting topics in geometry, it is clearly written and should be accessible to its target audience of students who have “a year of calculus, an introductory course in linear algebra, and an interest in mathematics”, and it offers students an opportunity to see the relationship between geometry and other areas of mathematics, including abstract and linear algebra. If you are teaching a course in geometry at the college level and the topics discussed here are compatible with your course, this book certainly merits a serious look.

Mark Hunacek ( teaches mathematics at Iowa State University.

See the table of contents in the publisher's webpage.